Closed-form Fisher–Rao distance for multivariate normal distributions

Derive a closed-form expression for the Fisher–Rao geodesic distance between two multivariate normal distributions with arbitrary mean vectors and covariance matrices, extending recent results that provide the geodesics with boundary value conditions.

Background

The paper emphasizes that, despite extensive paper, the Fisher–Rao distance is not known in closed form for multivariate normal distributions. Recent progress has produced explicit geodesics with boundary conditions, but integrating the Fisher–Rao length element along these geodesics to obtain a closed-form distance remains unresolved. Establishing such a formula would have significant implications across statistics, information geometry, and machine learning, where multivariate normals are ubiquitous.